A characterization of Leonard pairs using the parameters

Let V denote a vector space with finite positive dimension. We consider an ordered pair of linear transformations A:V→V and A*:V→V that satisfy (i) and (ii) below:(i)There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A* is diagonal.(ii)There exists a basis for V with respect to which the matrix representing A* is irreducible tridiagonal and the matrix representing A is diagonal.We call such a pair a Leonard pair on V. Pascasio recently obtained a characterization of the Q-polynomial distance-regular graphs using the intersection numbers ai. In this paper, we extend her results to a linear algebraic level and obtain a characterization of Leonard pairs. Pascasio’s argument appears to rely on the underlying combinatorial assumptions, so we take a different approach that is algebraic in nature.